We consider the system if.

  Alphabet:

    0 : [] --> nat 
    false : [] --> bool 
    h : [nat -> bool * nat -> nat * nat] --> nat 
    if : [bool * nat * nat] --> nat 
    s : [nat] --> nat 
    true : [] --> bool 

  Rules:

    if(true, x, y) => x 
    if(false, x, y) => y 
    h(f, g, 0) => 0 
    h(f, g, s(x)) => g h(f, g, if(f x, x, 0)) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [FuhKop19]).

We thus obtain the following dependency pair problem (P_0, R_0, computable, formative):

  Dependency Pairs P_0:

    0] h#(F, G, s(X)) =#> h#(F, G, if(F X, X, 0))   
    1] h#(F, G, s(X)) =#> if#(F X, X, 0)   

  Rules R_0:

    if(true, X, Y) => X 
    if(false, X, Y) => Y 
    h(F, G, 0) => 0 
    h(F, G, s(X)) => G h(F, G, if(F X, X, 0)) 

Thus, the original system is terminating if (P_0, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_0, R_0, computable, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 0, 1 
    * 1 :  

This graph has the following strongly connected components:

  P_1:

    h#(F, G, s(X)) =#> h#(F, G, if(F X, X, 0))   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f).

Thus, the original system is terminating if (P_1, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_1, R_0, computable, formative).

The formative rules of (P_1, R_0) are R_1 ::=

  if(true, X, Y) => X 
  if(false, X, Y) => Y 
  h(F, G, s(X)) => G h(F, G, if(F X, X, 0)) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, computable, formative) by (P_1, R_1, computable, formative).

Thus, the original system is terminating if (P_1, R_1, computable, formative) is finite.

We consider the dependency pair problem (P_1, R_1, computable, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  h#(F, G, s(X)) >? h#(F, G, if(F X, X, 0)) 
  if(true, X, Y) >= X 
  if(false, X, Y) >= Y 
  h(F, G, s(X)) >= G h(F, G, if(F X, X, 0)) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[@_{o -> o}(x_1, x_2)]] = @_{o -> o}(x_2, x_1) 
  [[h(x_1, x_2, x_3)]] = h(x_2, x_3) 
  [[h#(x_1, x_2, x_3)]] = h#(x_3, x_1) 
  [[if(x_1, x_2, x_3)]] = if(x_2, x_3) 

We choose Lex = {@_{o -> o}, h, h#, s} and Mul = {false, if, true}, and the following precedence: h# > true > h > @_{o -> o} = s > if > false

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  h#(F, G, s(X)) > h#(F, G, if(X, _|_)) 
  if(X, Y) >= X 
  if(X, Y) >= Y 
  h(F, G, s(X)) >= @_{o -> o}(G, h(F, G, if(X, _|_))) 

With these choices, we have:

  1] h#(F, G, s(X)) > h#(F, G, if(X, _|_))  because [2], by definition 
  2] h#*(F, G, s(X)) >= h#(F, G, if(X, _|_))  because [3], [8] and [10], by (Stat) 
  3] s(X) > if(X, _|_)  because [4], by definition 
  4] s*(X) >= if(X, _|_)  because s > if, [5] and [7], by (Copy) 
  5] s*(X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 
  7] s*(X) >= _|_  by (Bot) 
  8] h#*(F, G, s(X)) >= F  because [9], by (Select) 
  9] F >= F  by (Meta) 
  10] h#*(F, G, s(X)) >= if(X, _|_)  because [11], by (Select) 
  11] s(X) >= if(X, _|_)  because [4], by (Star) 

  12] if(X, Y) >= X  because [13], by (Star) 
  13] if*(X, Y) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 

  15] if(X, Y) >= Y  because [16], by (Star) 
  16] if*(X, Y) >= Y  because [17], by (Select) 
  17] Y >= Y  by (Meta) 

  18] h(F, G, s(X)) >= @_{o -> o}(G, h(F, G, if(X, _|_)))  because [19], by (Star) 
  19] h*(F, G, s(X)) >= @_{o -> o}(G, h(F, G, if(X, _|_)))  because h > @_{o -> o}, [20] and [22], by (Copy) 
  20] h*(F, G, s(X)) >= G  because [21], by (Select) 
  21] G >= G  by (Meta) 
  22] h*(F, G, s(X)) >= h(F, G, if(X, _|_))  because [23], [3], [20] and [24], by (Stat) 
  23] G >= G  by (Meta) 
  24] h*(F, G, s(X)) >= if(X, _|_)  because [11], by (Select) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_1) by ({}, R_1).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.